Existence of positive solutions for the fourth-order elliptic boundary value problems

This paper is concerned with the existence of a positive solution of the nonlinear fourth-order elliptic boundary value problem {Delta(2)u=f(x,u,Delta u), x is an element of Omega, u=Delta u=0, x is an element of partial derivative Omega, where Omega is a bounded smooth domain in R-N, f:Omega(over bar)xR+xR--> R+ is a continuous function. Under two inequality conditions of f(x,xi,eta) when |(xi,eta)| is small and large, an existence result of positive solutions is obtained. The inequality conditions is related to the principal eigenvalue lambda 1 of the Laplace operator -Delta with the boundary condition u|(partial derivative Omega)=0. The discussion is based on the fixed-point index theory in cones.